I don't quite understand the following simplification. I'm having a hard time wrapping my mind around exactly why this works. Is there some kind of simple example that someone could give me to make it "click"? Is there some sort of "rule" I can apply in similar situations?
$$(k+1)!\cdot (k+2)-1 = (k+2)!-1$$
Hint:
The factorial is defined recursively as:
$0!=1 \qquad (n+1)!=n!\cdot (n+1)$
so $$ (k+2)!=(k+1)!(k+2) $$ by definition.